Related papers: Spectral parameter power series for arbitrary orde…
Given a first-order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial…
A slightly modified variant of the cubic periodic one-dimensional nonlinear Schroedinger equation is shown to admit weak solutions for all initial data in certain function spaces wider than L^2. These solutions depend uniformly continuously…
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2{\alpha}.
Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonances, and fluid stability. Similarly, spectral decompositions (pure point, absolutely continuous and singular continuous) often…
We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schr\"odinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of…
The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = \lambda A(x)\mathbf{y}, \quad \ x \in [0, 1], $$ where $A(x) = diag\{a_1(x),…
A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of…
The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The…
Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system…
Complex-valued functions defined on a finite interval $[a,b]$ generalizing power functions of the type $(x-x_0)^n$ for $n\geq 0$ are studied. These functions called $\Phi$-generalized powers, $\Phi$ being a given nonzero complex-valued…
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
A classical theorem of Mihlin yields Lp estimates for spectral multipliers Lp(R^d) -> Lp(R^d); g -> F^{-1}[f(| |^2) Fg] in terms of L^\infty bounds of the multiplier function f and its weighted derivatives up to an order > d/2. This…
The Residual Power Series Method (RPSM) provides a powerful framework for solving fractional differential equations. However, a significant computational bottleneck arises from the necessity of calculating the fractional derivatives of the…
In this paper, the linear differential expression of order $n \ge 2$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we…
This paper is concerned with inverse spectral problems for higher-order ($n > 2$) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either…
We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary…
This paper describes various approaches to modeling a random process with a given rational power spectral density. The main attention is paid to the spectral form of mathematical description, which allows one to obtain a relation for the…
In this work we present a power series method for solving ordinary and partial differential equations. To demonstrate our method we solve a system of ordinary differential equations describing the movement of a random walker on a…
Power Series Solution Method has been traditionally used to solve Ordinary and Partial Linear Differential Equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In…